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Cacti Whose Spread is Maximal.
- Source :
-
Graphs & Combinatorics . Jan2015, Vol. 31 Issue 1, p23-34. 12p. - Publication Year :
- 2015
-
Abstract
- For a simple graph G, the graph's spread s( G) is defined as the difference between the largest eigenvalue and the least eigenvalue of the graph's adjacency matrix, i.e. $${s(G)=\rho(G)-\lambda(G)}$$ . A connected graph G is a cactus if any two of its cycles have at most one common vertex. If all cycles of the cactus G have exactly one common vertex then it is called a bundle. Let $${{\mathcal C}(n,k)}$$ denote the class of cacti with n vertices and k cycles. In this paper, we determine a unique cactus whose spread is maximal among the cacti with n vertices and k cycles. We prove that the obtained graph is a bundle of a special form. Within the class $${{\mathcal C}(n,k)}$$ we also present a unique cactus whose least eigenvalue is minimal (Petrović et al. in Linear Algebra Appl 435:2357-2364, ) and show that these two graphs are the same, except for a few cases in which n is small. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 09110119
- Volume :
- 31
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Graphs & Combinatorics
- Publication Type :
- Academic Journal
- Accession number :
- 100084847
- Full Text :
- https://doi.org/10.1007/s00373-013-1373-1